The problem provided specifies:

• $a_2 = 12$
• $a_5 = 324$
• Find $a_1$.

This setup suggests an arithmetic or geometric sequence. However, given that the values grow significantly, it is likely a geometric sequence. For a geometric sequence, we use:

$a_n = a_1 \cdot r^{n-1}$

where $a_n$ is the $n$-th term, $a_1$ is the first term, and $r$ is the common ratio.

Step-by-Step Solution:

1. Using $a_2 = 12$: $a_2 = a_1 \cdot r = 12$

2. Using $a_5 = 324$: $a_5 = a_1 \cdot r^4 = 324$

3. Set up a system of equations: $a_1 \cdot r = 12$ $a_1 \cdot r^4 = 324$

4. Solve for $r$: Divide the second equation by the first equation: $\frac{a_1 \cdot r^4}{a_1 \cdot r} = \frac{324}{12}$ $r^3 = 27$ $r = 3$

5. Substitute $r = 3$ back into the first equation to find $a_1$: $a_1 \cdot 3 = 12$ $a_1 = \frac{12}{3} = 4$

Answer:

$a_1 = 4$

Would you like more details or have any questions?

Additional Questions:

1. How do we determine if a sequence is geometric or arithmetic?
2. Can we solve similar problems if only two terms are given?
3. What would change in our approach if this were an arithmetic sequence?
4. How can we verify the answer for $a_1$ by substituting values back?
5. What is the general formula for the $n$-th term in an arithmetic sequence?

Tip:

When solving for the first term in a geometric sequence, identifying the common ratio first often simplifies the process.